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| '''Hilbert's nineteenth problem''' is one of the 23 [[Hilbert problems]], set out in a celebrated list compiled in 1900 by [[David Hilbert]].<ref>See {{harv|Hilbert|1900}} or, equivalently, one of its translations.</ref> It asks whether the solutions of regular problems in the calculus of variations are always [[analytic function|analytic]].<ref>"''Sind die Lösungen regulärer Variationsprobleme stets notwending analytisch?''" (English translation by [[Mary Frances Winston Newson]]:-"''Are the solutions of regular problems in the calculus of variations always necessarily analytic?''"), formulating the problem with the same words of {{harvtxt|Hilbert|1900|p=288}}.</ref> Informally, and perhaps less directly, since Hilbert's concept of a "''regular variational problem''" identifies precisely a [[Calculus of variation|variational problem]] whose [[Euler-Lagrange equation]] is an [[elliptic partial differential equation]] with analytic coefficients,<ref>See {{harv|Hilbert|1900|pp=288–289}}, or the corresponding section on the nineteenth problem in any of its translation or reprint, or the subsection "[[Hilbert's nineteenth problem#The origins of the problem|The origins of the problem]]" in the historical section of this entry.</ref> Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of [[partial differential equation]]s, any solution function inherit the relatively simple and well understood structure from the solved equation.
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| ==History==
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| ===The origins of the problem===
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| {{quote
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| |text= Eine der begrifflich merkwürdigsten Thatsachen In den Elementen der Theorie der analytischen Funktionen erblicke ich darin, daß es Partielle Differentialgleichungen giebt, deren Integrale sämtlich notwendig analytische Funktionen der unabhängigen Variabeln sind, die also, kurz gesagt, nur analytischer Lösungen fähig sind.<ref>English translation by Mary Frances Winston Newson:-"''One of the most remarkable facts in the elements of the theory of analytic functions appears to me to be this: that there exist partial differential equations whose integrals are all of necessity analytic functions of the independent variables, that is, in short, equations susceptible of none but analytic solutions''".</ref>
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| |sign= [[David Hilbert]]
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| |source= {{harv|Hilbert|1900|p=288}}.
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| }}
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| David Hilbert presented the now called Hilbert's nineteen problem in his speech at the second [[International Congress of Mathematicians]].<ref>For a detailed historical analysis, see the relevant entry "[[Hilbert's problems]]".</ref> In {{harv|Hilbert|1900|p=288}} he states that, in his opinion, one of the most remarkable facts of the theory of analytic functions is that there exist classes of partial differential equations which admit only such kind of functions as solutions, adducing [[Laplace's equation]], [[Liouville's equation]],<ref>Hilbert does not cite explicitly [[Joseph Liouville]] and considers the constant [[Gaussian curvature]] {{math|''K''}} as equal to {{math|-1/2}}: compare the relevant entry with {{harv|Hilbert|1900|p=288}}.</ref> the [[minimal surface equation]] and a class of linear partial differential equations studied by [[Émile Picard]] as examples.<ref>Contrary to Liouville's work, Picard's work is explicitly cited by {{harvtxt|Hilbert|1900|loc=p. 288 and footnote 1 in the same page}}.</ref> He then notes the fact that most of the partial differential equations sharing this property are the Euler-Lagrange equation of a well defined kind of variational problem, featuring the following three properties:<ref name="Hilbertp288">See {{harv|Hilbert|1900|p=288}}.</ref>
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| :{{EquationRef|1|(1){{spaces|5}}}}<math>{\iint F(p,q,z;x,y) dx dy} = \text{Minimum} \qquad
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| \left[ \frac{\partial z}{\partial x}=p \quad;\quad \frac{\partial z}{\partial y}=q \right]</math>,
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| :{{EquationRef|2|(2){{spaces|5}}}}<math>\frac{\partial^2 F}{\partial^2 p}\cdot\frac{\partial^2 F}{\partial^2 q} - \left(\frac{\partial^2 F}{{\partial p}{\partial q}}\right)^2 > 0</math>,
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| :{{EquationRef|3|(3){{spaces|5}}}} {{math|''F''}} is an analytic function of all its arguments {{math|''p'', ''q'', ''z'', ''x''}} and {{math|''y''}}.
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| Hilbert calls this kind of variational problem a "''regular variational problem''":<ref>"''Reguläres Variationsproblem''", in his exact words. Hilbert's definition of a regular variational problem is stronger than the currently used one, found, for example, in {{harv|Gilbarg|Trudinger|2001|p=289}}.</ref> property {{EquationNote|(1)}} means that such kind of variational problems are [[minimum|minimum problems]], property {{EquationNote|(2)}} is the [[Elliptic partial differential equation|ellipticity condition]] on the Euler-Lagrange equations associated to the given [[Functional (mathematics)|functional]], while property {{EquationNote|(3)}} is a simple regularity assumption the function {{math|''F''}}.<ref>Since Hilbert considers all [[derivative]]s in the "classical", i.e. not in the [[Weak derivative|weak]] nor in the [[strong derivative|strong]], sense, even before the statement of its analyticity in {{EquationNote|(3)}}, the function {{math|''F''}} is assumed to be at least {{math|{{SubSup|C||2}}}}, as the use of the [[Hessian determinant]] in {{EquationNote|(2)}} implies.</ref> Having identified the class of problems to deal with, he then poses the following question:-"''... does every Lagrangian partial differential equation of a regular variation problem have the property of admitting analytic integrals exclusively?''"<ref>English translation by Mary Frances Winston Newson: [[#{{harvid|Hilbert|1900}}|Hilbert's (1900]], p. 288) precise words are:-"''... d. h. ob jede Lagrangesche partielle Differentialgleichung eines reguläres Variationsproblem die Eigenschaft at, daß sie nur analytische Integrale zuläßt''" ([[Italic type|Italics empahsis]] by Hilbert himself).</ref> and asks further if this is the case even when the function is required to assume, as it happens for Dirichlet's problem on the [[potential theory|potential function]], boundary values which are continuous, but not analytic.<ref name="Hilbertp288" />
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| ===The path to the complete solution===
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| Hilbert stated his nineteenth problem as a [[regularity problem]] for a class of elliptic partial differential equation with analytic coefficients,<ref name="Hilbertp288" /> therefore the first efforts of the researchers who sought to solve it were directed to study the regularity of [[classical solution]]s for equations belonging to this class. For [[Continuously differentiable function|{{math|{{SubSup|C||3}}}}]] solutions Hilbert's problem was answered positively by {{harvs|txt|first=Sergei|last=Bernstein|authorlink=Sergei Natanovich Bernstein|year=1904}} in his thesis: he showed that {{math|{{SubSup|C||3}}}} solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved over the years by several authors, such as {{harvtxt|Petrowsky|1939}}, who reduced the differentiability requirements on the solution needed to prove that it is analytic. On the other hand, direct methods in the calculus of variations showed the existence of solutions with very weak differentiability properties. For many years there was a gap between these results: the solutions that could be constructed were known to have square integrable second derivatives, which was not quite strong enough to feed into the machinery that could prove they were analytic, which needed continuity of first derivatives. This gap was filled independently by {{harvs|txt|author-link=Ennio de Giorgi|first=Ennio |last= De Giorgi|year1=1956|year2= 1957}}, and {{harvs|txt|first=John Forbes |last=Nash|author-link=John Forbes Nash|year1=1957|year2=1958}}. They were able to show the solutions had first derivatives that were [[Hölder continuous]], which by previous results implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem.
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| ===Counterexamples to various generalizations of the problem===
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| The affirmative answer to Hilbert's nineteenth problem given by Ennio De Giorgi and John Forbes Nash raised the question if the same conclusion holds also for Euler-lagrange equations of more general [[Functional (mathematics)|functional]]s: at the end of the [[sixties]], {{harvtxt|Maz'ya|1968}},<ref>See {{harv|Giaquinta|1983|p=59}}, {{harv|Giusti|1994|loc=p. 7 footnote 7 and p. 353}}, {{harv|Gohberg|1999|p=1}}, {{harv|Hedberg|1999|pp=10–11}}, {{harv|Kristensen|Mingione|2011|loc=p. 5 and p. 8}}, and {{harv|Mingione|2006|p=368}}.</ref> {{harvtxt|De Giorgi|1968}} and {{harvtxt|Giusti|Miranda|1968}} constructed independently several [[counterexample]]s,<ref>See {{harv|Giaquinta|1983|pp=54–59}}, {{harv|Giusti|1994|loc=p. 7 and pp. 353}}.</ref> showing that in general there is no hope to prove such kind of regularity results without adding further hypotheses.
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| Precisely, {{harvtxt|Maz'ya|1968}} gave several counterexamples involving a single elliptic equation of order greater than two with analytic coefficients:<ref>See {{harv|Hedberg|1999|pp=10–11}}, {{harv|Kristensen|Mingione|2011|loc=p. 5 and p. 8}} and {{harv|Mingione|2006|p=368}}.</ref> for experts, the fact that such kind of equations could have nonanalytic and even nonsmooth solutions created a sensation.<ref>According to {{harv|Gohberg|1999|p=1}}.</ref>
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| {{harvtxt|De Giorgi|1968}} and {{harvtxt|Giusti|Miranda|1968}} gave counterexamples showing that in the case when the solution is vector-valued rather than scalar-valued, it need not to be analytic: the example of De Giorgi consists of an elliptic system with bounded coefficients, while the one of Giusti and Miranda has analytic coefficients.<ref>See {{harv|Giaquinta|1983|pp=54–59}} and {{harv|Giusti|1994|loc=p. 7, pp. 202–203 and pp. 317–318}}.</ref> Later on, {{harvtxt|Nečas|1977}} provided other, more refined, examples for the vector valued problem.<ref>For more information about the work of [[Jindřich Nečas]] see the work of {{harvtxt|Kristensen|Mingione|2011|loc=§3.3, pp. 9–12}} and {{harv|Mingione|2006|loc=§3.3, pp. 369–370}}.</ref>
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| ==De Giorgi's theorem==
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| The key theorem proved by De Giorgi is an [[a priori estimate]] stating that if ''u'' is a solution of a suitable linear second order strictly elliptic PDE of the form
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| :<math> D_i(a^{ij}(x)D_ju)=0</math>
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| and ''u'' has square integrable first derivatives, then ''u'' is Hölder continuous.
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| ==Application of De Giorgi's theorem to Hilbert's problem==
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| Hilbert's problem asks whether the minimizers ''w'' of an energy functional such as
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| :<math>\int_UL(Dw)\mathrm{d}x</math>
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| are analytic. Here ''w'' is a function on some compact set ''U'' of '''R'''<sup>''n''</sup>, ''Dw'' is its [[gradient]] vector, and ''L'' is the Lagrangian, a function of the derivatives of ''w'' that satisfies certain growth, smoothness, and convexity conditions. The smoothness of ''w'' can be shown using De Giorgi's theorem
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| as follows. The Euler-Lagrange equation for this variational problem is the non-linear equation
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| :<math> \Sigma_i(L_{p_i}(Dw))_{x_i} = 0</math>
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| and differentiating this with respect to ''x''<sub>''k''</sub> gives
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| :<math> \Sigma_i(L_{p_ip_j}(Dw)w_{x_jx_k})_{x_i} = 0</math>
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| This means that ''u''=''w''<sub>''x''<sub>''k''</sub></sub> satisfies the linear equation
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| :<math> D_i(a^{ij}(x)D_ju)=0</math>
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| with
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| :<math>a^{ij} = L_{p_ip_j}(Dw)</math>
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| so by De Giorgi's result the solution ''w'' has Hölder continuous first derivatives.
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| Once ''w'' is known to have Hölder continuous (''n''+1)st derivatives for some ''n'' ≥ 0, then the coefficients ''a''<sup>''ij''</sup> have Hölder continuous ''n''th derivatives, so a theorem of Schauder implies that the (''n''+2)nd derivatives are also Hölder continuous, so repeating this infinitely often shows that the solution ''w'' is smooth.
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| ==Nash's theorem==
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| Nash gave a continuity estimate for solutions of the parabolic equation
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| :<math> D_i(a^{ij}(x)D_ju)=D_t(u)</math>
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| where ''u'' is a bounded function of ''x''<sub>1</sub>,...,''x''<sub>''n''</sub>, ''t'' defined for ''t'' ≥ 0. From his estimate Nash was able to deduce a continuity estimate for solutions of the elliptic equation
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| :<math> D_i(a^{ij}(x)D_ju)=0</math> by considering the special case when ''u'' does not depend on ''t''.
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| ==Notes==
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| {{reflist|29em}}
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| }}. "''On the analiticity of extremals of multiple integrals''" (English translation of the title) is a short research announcement disclosing the results detailed later in {{harv|De Giorgi|1957}}. While, according to the [[#{{harvid|De Giorgi|2006}}|Complete list of De Giorgi's scientific publication (De Giorgi 2006]], p. 6), an English translation should be included in {{harv|De Giorgi|2006}}, it is unfortunately missing.
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| }}. Translated in English as "''On the differentiability and the analiticity of extremals of regular multiple integrals''" in {{harv|De Giorgi|2006|pp=149–166}}.
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| }}. Translated in English as "''An example of discontinuous extremals for a variational problem of elliptic type''" in {{harv|De Giorgi|2006|pp=285–287}}.
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| | pages = 253–297
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| | year = 1900
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| | language = German
| |
| | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1900&DMDID=DMDLOG_0037
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| | jfm =31.0068.03
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| }} (reprinted as {{Citation
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| | title = Mathematische Probleme
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| | journal = [[Archiv der Mathematik und Physik]]
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| | series = dritte reihe
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| | volume = 1
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| | pages = 44–63 and 253–297
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| | year = 1900
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| | language = German
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| | url = https://archive.org/stream/archivdermathem02unkngoog#page/n61/mode/1up
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| }} (reprinted as {{Citation
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| | last = Hilbert
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| }}), and in French (with additions of Hilbert himself) by M. L. Laugel as {{Citation
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| | title = Compte Rendu du Deuxième Congrès International des Mathématiciens, tenu à Paris du 6 au 12 août 1900. Procès-Verbaux et Communications
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| | series = ICM Proceedings
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| | title = Sketches of Regularity Theory from The 20th Century and the Work of Jindřich Nečas
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| | volume = Report no. OxPDE-11/17
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| | place = Oxford
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| |date=October 2011
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| | url = http://www.maths.ox.ac.uk/system/files/attachments/OxPDE_11-17.pdf
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| | url = http://mi.mathnet.ru/eng/faa/v2/i3/p53
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| }}, translated in English as {{Citation
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| | title = Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients
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| | editor-last =Kluge
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| | editor-first =Reinhard
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| | editor2-last =Müller
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| | editor2-first =Wolfdietrich
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| | contribution = Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity
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| | contribution-url =
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| | title = Theory of nonlinear operators: constructive aspects. Proceedings of the fourth international summer school, held at Berlin, GDR, from September 22 to 26, 1975
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| | series = Abhandlungen der Akademie der Wissenschaften der DDR
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| | volume = Nr. 1N
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| | publisher = [[Akademie-Verlag]]
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| *{{Citation
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| | title= Sur l'analyticité des solutions des systèmes d'équations différentielles
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| | url= http://mi.mathnet.ru/eng/msb5769
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| | year=1939
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| | language = French
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| | journal = [[Matematicheskii Sbornik|Recueil Mathématique (Matematicheskii Sbornik)]]
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| | volume = 5(47)
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| | issue = 1
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| | pages = 3–70
| |
| | jfm = 65.0405.02
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| | mr = 0001425
| |
| | zbl = 0022.22601
| |
| }}.
| |
| | |
| {{Hilbert's problems}}
| |
| | |
| [[Category:Hilbert's problems|#19]]
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Calculus of variations]]
| |